When we return to Einstein's derivations of the transform
formulas with a well-focused eye, we find he was a wee bit
confused - or at least self-contradictory.
When he set up his (at first unknown) tau=moving system
time formulas, he created three particular instances of tau.
Tau.0 is the time at which light is emitted at the moving
origin toward a mirror to the right that is moving at rest
wrt that moving origin and at a constant distance from that
origin. He lets the stationary time slot have the value t,
a constant, the stationary system starting time.
Tau.1 is the time at which the light is reflected. He
lets the stationary time be t+x'/(c-v); t is still a
constant and x'/(c-v) is the time interval since t.
Tau.2 is the time at which the light gets (back) to the
moving origin. The stationary time value is put as t +
x'/(c-v) + x'/(c+v); t is still a constant and x'/(c-v)
+ x'/(c+v) is the time interval since t.
On the thesis that the moving observer sees the time to
the mirror as the same as the time back to the origin,
he sets
.5[ tau.0 + tau.2 ] = tau.1.
Tau.0 completely drops out of the analysis and leaves
no trace, and has no effect.
Further, the t you see in tau.0, tau.1, and tau.2 also
completely drops out with no trace and no effect, leaving
us with exactly what you'd get if you had explicilty said
t' is an interval and so is t.
What doesn't drop out in the stationary time values is
x'/(c-v) and x'/(c+v), the time interval it takes for
light to get to the fleeing mirror, and the time interval
it takes for light to get back to the approaching origin.
Thus, his resultant t' formula is strictly based on time
intervals in the stationary system. Time intervals since
some starting time, yes, but time intervals.
There is absolutely nothing in the derived formulas that
depends on arbitrary coordinates like the constant t in
the stationary time arguments.
Let's look at the x dimension; it is x'=x-vt [as x increases
by vt, the effect over time is x'=(x+vt)-vt)], which Einstein
explicitly sets up as a constant stationary distance.
He uses that x' not just in the time interval parts of the
stationary time arguments, but also in the x (distance)
stationary system argument for the tau at the time light
is reflected.
x' can't be the stationary system coordinate of the mirror
at that time. That value is x'+vt.
x' is explicitly an interval, distance.
Thus, the whole tau derivation of the t' formula is fully and
explicitly based on x' - a spatial length/distance/interval -
and the two time interals x'/(c-v) and x'/(c+v).
While we're at it, if the starting t is not zero, his
x'=x-vt formula is complete nonsense also. Given that
there was some L that was the mirror x-location and length
when the light is emitted, if t was already, say, 500, then
x'=L-vt could have been a very negative length.

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